266,508 research outputs found
The number of conformally equivalent maximal graphs
We show that the number of entire maximal graphs with finitely many singular
points that are conformally equivalent is a universal constant that depends only
on the number of singularities, namely 2n for graphs with n + 1 singularities. We
also give an explicit description of the family of entire maximal graphs with a finite
number of singularities all of them lying on a plane orthogonal to the limit normal
vector at infinity.Ministerio de Educación y Ciencia MTM2007-64504Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
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